In the next section, we put down formulations of the quantum spatial
translations and the key relevant results following from Refs.[1]}, [2]},
to set the platform for our analysis. Note that our explicit formulation
has a part, as the representation of the states in the full Hilbert space
including the old and new reference frames, that is different from that
of Ref.[2]}. We will discuss the merit of the formulation as an
improvement on the latter.
Sec., is then devoted to presenting and analyzing the
changes in the noncommutative values of the observables under
the quantum spatial translations for a couple of illustrative cases.
Most readers are probably new to the notion of the noncommutative
values. In the beginning of the section, we give an essentially
self-contained presentation of a convenient new variant of it. The
part together with the explicit functional representation under the
use of Schrödinger wavefunction presented in the Appendix
should be enough for following and understanding our analyses.
The fact that the notion be conceptually new, however, means that
one needs to be careful to follow the exact mathematical logic
involved to avoid misinterpreting results, and may have to bear
with the uncomfortable feeling of dealing with something unfamiliar.
We have to beg the readers patience on that.
